123 
effect in changing the sign of A as the interchange of the variables alone. 
Hence when all the factors are commutative, the A may operate either on the 
variables or on the functional symbols, and since A ¢ is linear in the latter, it has 
the properties of linear alternates with respect to the functional symbols. If the 
functional symbols be linear, the alternate A ¢ is also linear with respect to the 
variables. 
VI. QUATERNIONS. 
29. We consider linear alternate products whose functional symbols are 1, S, 
V, K. The symbol S gives a factor that is commutative with any other factor, so 
that any other symbol in the same product with S may be reduced by +17 S, 
where n is a scalar. | 
30. By substituting 1— S-++ V, K—S— V and expanding, our linear 
alternate product of any order is found to depend on two in which the symbols 
are either all V or one S and the rest V. Two S symbols give an alternate pro- 
duct that is identically zero (Theor. 2, note). It appears that: the two of second 
order are vectors; the two of third order are a scalar and a vector; one of the 
fourth order is zero, the’other is a scalar. Any linear alternate of fifth or higher 
order is identically zero. 
31. In the geometrical interpretation in which 1, 7, 7, k are the numbers of 
four mutually perpendicular unit lines in four-fold space, the condition of per- 
pendicularity of p, gisS.pKq=o=S.Kp.qie,pKq——qKp, Kp.4q 
—=—Kq.p. Thus in any alternate product whose functional symbols are alter- 
nately 1, K, and whose variables occur in sets, such that any two of different sets 
are perpendicular, we have 
ap =A A, D, 
where A’ A” are alternate symbols that affect the different sets of variables [th 27 
(b), art. 16.] In particular, if all the variables are mutually perpendicular, then 
aot A“ — 1, and A.¢d= 9. 
32. The alternates of second order are: , 
ale”, Ae po g—= VV, Vp. Vig bd FON ik. 
ba2 a. SpeVig- = 24.8 p.q—Ai-+ By + Ck. 
A, B, OC, L, M, N are the six independent scalar linear alter- 
nates of second order, and are the coefficients of © (1, 7), 
@ (1, j), 9 (1, k), > (j,k), @ (&, 7), 9 (4, J), in the expansion of 
any linear alternate © (p, q). 
